Paradoxes of C o m m o n K n o w l e d g e Revisited:
A P e r s p e c t i v e from G a m e T h e o r y and E c o n o m i c s
Abstract
Stephen Morris
Department of Economics
University of Pennsylvania
3718 Locust Walk, Philadelphia PA 19104
Email: [email protected]
Common knowledge and thus perfect coordination are impossible in any
information system where there are communication errors. This is true however unlikely the communication errors and however long and detailed the
communication. This is the canonical example of a "paradox" of common
knowledge; such paradoxes have been studied in recent years by researchers
in computer science, philosophy, economics and other fields. The study of
such paradoxes served two important functions. First, researchers attempting to account for them developed formal ways of thinking about interactive
knowledge. Second, because the same paradoxes attracted attention across
many fields, they forced (encouraged?!) researchers in the different fields to
talk to each other.
But granted that these two functions were successfully performed, do we
have anything more to learn from the paradoxes? In practical problems,
something less than common knowledge ("almost common knowledge") is
often enough to achieve co-ordination. Is the extreme case of common knowledge instructive about problems of co-ordination or is an intellectual curiosum with few implications for practical problems of co-ordination? These
questions are the subject of this discussion session. I shall answer them with
a qualified yes.
Before doing so, I want to note how an economist's perspective on these
problems will differ from others' perspectives because of strategic issues.
First, a physical description of a communication protocol is not sufficient
for the economist. It is important to also check that communicators have
an incentive to tell the truth. Second, even granted honest communication,
the relevant notion of almost common knowledge is going to depend on the
kind of co-ordination problem studied, and thus the field. Let me illustrate
this with the co-ordinated attack problem. Consider the version where a
first general sends an instruction to attack to a second; confirmations are
sent back and forth until one gets lost (each message is lost with some small
probability), and the two generals must attack simultaneously at dawn. It
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is impossible to describe a rule such that (1) attack occurs with positive
probability and (2) it is never the case that one general attacks alone. The
economist, on the other hand, would typically be perfectly happy to ensure
that it is exceedingly unlikely that one general attacks alone. A simple rule
which ensures this is for the first general to always attack and the second
general to attack if he has received at least one message. But can the generals be relied on to follow such an instruction? Suppose that the generals do
not necessarily follows orders, and that each general's preferences are such
that he would not want to attack unless he attached probability significantly
greater than half to the other general attacking. Then this simple rule would
not work. If the first general had received no confirmation, he would attach
probability about one half to his first message never having arrived. Since
the second general would not attack in that contingency, the first general will
not attack. A similar argument by contradiction can be used to show that no
attack ever happens in any equilibrium of this incomplete information game.
Thus even if the analyst is prepared to accept a small probability of failure of
co-ordination, this small probability may become large because of strategic
concerns. Since processors presumably do not have different preferences, this
issue does not arise in computer science.
Subject to these two provisos, economists should join with other researchers in answering the following questions:
1. Given that perfect co-ordination is not possible with imperfect communication, what is the best that can be done subject to a given information system / communication protocol?
2. To the extent that information systems / communication protocols can
be designed (and - for the economist - made incentive compatible), how
should they be designed?
I will report on some work in economics addressing these questions and make
suggestions for future work.
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